We denote \(\mathcal {H}\)  as the Hilbert space (over the set of real numbers \(\mathbb {R}\) ) and \(\mathcal {H}^*\)  as its dual space. We inherit Dirac's bra-ket notations [1]} to denote generic quantum states: \(\langle \varphi |\in \mathcal {H}^*,|\varphi \rangle \in \mathcal {H}\) . Let \(B(\mathcal {H})\)  be the space of all positive, Hermitian and bounded operators from \(\mathcal {H}\)  to itself. Let \(\mathcal {S}\)  be the subset of \(B(\mathcal {H})\)  such trace of its operators is 1.
In addition, we denote \(\mathcal {V}\)  as the space of projection-valued measurements [2]}.
We denote \(\mathbf {1}\in B(\mathcal {H})\)  as the identity operator. For any operator \(A\in B(\mathcal {H})\) , we denote its conjugate transpose as \(A^{\dagger }\) .
